.. SPDX-FileCopyrightText: 2019-2020 Intel Corporation .. .. SPDX-License-Identifier: CC-BY-4.0 .. _onemkl_lapack_potrs: potrs ===== Solves a system of linear equations with a Cholesky-factored symmetric (Hermitian) positive-definite coefficient matrix. .. container:: section .. rubric:: Description potrs supports the following precisions. .. list-table:: :header-rows: 1 * - T * - float * - double * - std::complex * - std::complex The routine solves for :math:X the system of linear equations :math:AX = B with a symmetric positive-definite or, for complex data, Hermitian positive-definite matrix :math:A, given the Cholesky factorization of :math:A: .. list-table:: :header-rows: 1 * - :math:A = U^TU for real data, :math:A = U^HU for complex data - if upper_lower=oneapi::mkl::uplo::upper * - :math:A = LL^T for real data, :math:A = LL^H for complex data - if upper_lower=oneapi::mkl::uplo::lower where :math:L is a lower triangular matrix and :math:U is upper triangular. The system is solved with multiple right-hand sides stored in the columns of the matrix :math:B. Before calling this routine, you must call :ref:onemkl_lapack_potrf to compute the Cholesky factorization of :math:A. potrs (Buffer Version) ---------------------- .. container:: section .. rubric:: Syntax .. code-block:: cpp namespace oneapi::mkl::lapack { void potrs(cl::sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, std::int64_t nrhs, cl::sycl::buffer &a, std::int64_t lda, cl::sycl::buffer &b, std::int64_t ldb, cl::sycl::buffer &scratchpad, std::int64_t scratchpad_size) } .. container:: section .. rubric:: Input Parameters queue The queue where the routine should be executed. upper_lower Indicates how the input matrix has been factored: If upper_lower = oneapi::mkl::uplo::upper, the upper triangle :math:U of :math:A is stored, where :math:A = :math:U^{T}U for real data, :math:A = :math:U^{H}U for complex data. If upper_lower = oneapi::mkl::uplo::lower, the lower triangle :math:L of :math:A is stored, where :math:A = :math:LL^{T} for real data, :math:A = :math:LL^{H} for complex data. n The order of matrix :math:A (:math:0 \le n).\ nrhs The number of right-hand sides (:math:0 \le \text{nrhs}). a Buffer containing the factorization of the matrix A, as returned by :ref:onemkl_lapack_potrf. The second dimension of a must be at least :math:\max(1, n). lda The leading dimension of a. b The array b contains the matrix :math:B whose columns are the right-hand sides for the systems of equations. The second dimension of b must be at least :math:\max(1,\text{nrhs}). ldb The leading dimension of b. scratchpad_size Size of scratchpad memory as a number of floating point elements of type T. Size should not be less than the value returned by :ref:onemkl_lapack_potrs_scratchpad_size function. .. container:: section .. rubric:: Output Parameters b Overwritten by the solution matrix :math:X. scratchpad Buffer holding scratchpad memory to be used by routine for storing intermediate results. .. container:: section .. rubric:: Throws This routine shall throw the following exceptions if the associated condition is detected. An implementation may throw additional implementation-specific exception(s) in case of error conditions not covered here. :ref:oneapi::mkl::host_bad_alloc :ref:oneapi::mkl::device_bad_alloc :ref:oneapi::mkl::unimplemented :ref:oneapi::mkl::unsupported_device :ref:oneapi::mkl::lapack::invalid_argument :ref:oneapi::mkl::lapack::computation_error Exception is thrown in case of problems during calculations. The info code of the problem can be obtained by info() method of exception object: If :math:\text{info}=-i, the :math:i-th parameter had an illegal value. If :math:\text{info}=i, the :math:i-th diagonal element of the Cholesky factor is zero, and the solve could not be completed. If info equals to value passed as scratchpad size, and detail() returns non zero, then passed scratchpad is of insufficient size, and required size should not be less than value return by detail() method of exception object. potrs (USM Version) ---------------------- .. container:: section .. rubric:: Syntax .. code-block:: cpp namespace oneapi::mkl::lapack { cl::sycl::event potrs(cl::sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, std::int64_t nrhs, T *a, std::int64_t lda, T *b, std::int64_t ldb, T *scratchpad, std::int64_t scratchpad_size, const std::vector &events = {}) } .. container:: section .. rubric:: Input Parameters queue The queue where the routine should be executed. upper_lower Indicates how the input matrix has been factored: If upper_lower = oneapi::mkl::uplo::upper, the upper triangle :math:U of :math:A is stored, where :math:A = :math:U^{T}U for real data, :math:A = :math:U^{H}U for complex data. If upper_lower = oneapi::mkl::uplo::lower, the lower triangle :math:L of :math:A is stored, where :math:A = :math:LL^{T} for real data, :math:A = :math:LL^{H} for complex data. n The order of matrix :math:A (:math:0 \le n).\ nrhs The number of right-hand sides (:math:0 \le \text{nrhs}). a Pointer to array containing the factorization of the matrix :math:A, as returned by :ref:onemkl_lapack_potrf. The second dimension of a must be at least :math:\max(1, n). lda The leading dimension of a. b The array b contains the matrix :math:B whose columns are the right-hand sides for the systems of equations. The second dimension of b must be at least :math:\max(1,\text{nrhs}). ldb The leading dimension of b. scratchpad_size Size of scratchpad memory as a number of floating point elements of type T. Size should not be less than the value returned by :ref:onemkl_lapack_potrs_scratchpad_size function. events List of events to wait for before starting computation. Defaults to empty list. .. container:: section .. rubric:: Output Parameters b Overwritten by the solution matrix :math:X. scratchpad Pointer to scratchpad memory to be used by routine for storing intermediate results. .. container:: section .. rubric:: Throws This routine shall throw the following exceptions if the associated condition is detected. An implementation may throw additional implementation-specific exception(s) in case of error conditions not covered here. :ref:oneapi::mkl::host_bad_alloc :ref:oneapi::mkl::device_bad_alloc :ref:oneapi::mkl::unimplemented :ref:oneapi::mkl::unsupported_device :ref:oneapi::mkl::lapack::invalid_argument :ref:oneapi::mkl::lapack::computation_error Exception is thrown in case of problems during calculations. The info code of the problem can be obtained by info() method of exception object: If :math:\text{info}=-i, the :math:i-th parameter had an illegal value. If :math:\text{info}=i, the :math:i-th diagonal element of the Cholesky factor is zero, and the solve could not be completed. If info equals to value passed as scratchpad size, and detail() returns non zero, then passed scratchpad is of insufficient size, and required size should not be less than value return by detail() method of exception object. .. container:: section .. rubric:: Return Values Output event to wait on to ensure computation is complete. **Parent topic:** :ref:onemkl_lapack-linear-equation-routines`